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\title{The restriction problem and the Frobenius transform}

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\author{\texorpdfstring{Mitchell Lee\thanks{\href{mailto:mitchell@math.harvard.edu}{mitchell@math.harvard.edu}}\addressmark{1}}{Mitchell Lee}}

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\address{\addressmark{1}Department of Mathematics, Harvard University}

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\abstract{We define an abelian group homomorphism~$\mathscr{F}$, which we call the \emph{Frobenius transform}, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of~$\mathscr{F}$ in the Schur basis are the \emph{restriction coefficients} $r_\lambda^\mu = \dim \operatorname{Hom}_{\mathfrak{S}_n}(V_\mu, \mathbb{S}^\lambda \mathbb{C}^n)$, which are known to be nonnegative integers but have no known combinatorial interpretation.

We compute $\mathscr{F}\{f\}$ when $f$ is an elementary, complete homogeneous, or power sum symmetric function. As a consequence, we prove that $r_\lambda^\mu = 0$ if $|\lambda \cap \hat{\mu}| < 2|\hat{\mu}| - |\lambda|$, where $\hat{\mu}$ is the partition formed by removing the first part of~$\mu$. We also prove that $r_\lambda^\mu = 0$ if the Young diagram of~$\mu$ contains a square of side length greater than $2^{\lambda_1 - 1}$, and this inequality is tight.}

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\keywords{symmetric functions, representation theory of categories, plethysm, Kronecker product}

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\begin{document}

\maketitle
\section{Introduction}
Let $n \geq 0$ and let~$\lambda$ be a partition with at most~$n$ parts. There is a corresponding irreducible $GL_n(\K)$-module: the Schur module $\mathbb{S}^\lambda \K^n$. Because the symmetric group $\Sym_n$ embeds in $GL_n(\K)$ by permutation matrices, one may ask: how does the restriction of $\mathbb{S}^\lambda \K^n$ to $\Sym_n$ decompose into irreducible $\Sym_n$-modules?

In other words, let~$\lambda$ and~$\mu$ be partitions and let $n = |\mu|$. What is the value of the \textit{restriction coefficient} \[r_{\lambda}^\mu = \dim \operatorname{Hom}_{\Sym_n}(V_\mu, \mathbb{S}^\lambda \K^n),\] where $V_\mu$ is the Specht module corresponding to the partition~$\mu$? This problem is known as the \emph{restriction problem}.\footnote{The restriction problem is a special case of the more general problem of computing the \emph{inner plethysm coefficients}, which were defined by Littlewood in 1958 \cite{MR95208}.}

While there are many known formulas for the restriction coefficient $r_\lambda^\mu$, no \emph{combinatorial} formula is known. For example, in 1935, Littlewood proved a plethystic formula for restriction coefficients \cite{MR1576896}: \[r_{\lambda}^\mu = \langle s_\lambda, s_\mu[1 + h_1 + h_2 + \cdots] \rangle,\] where $s_\lambda$ denotes the Schur function, $h_r$ denotes the complete homogeneous symmetric function, $\langle \cdot, \cdot\rangle$ denotes the Hall inner product, and the square brackets denote plethysm of symmetric functions. However, this formula does not obviously yield a combinatorial interpretation of $r_\lambda^\mu$.

The restriction problem has attracted a great deal of interest. Here is a sampling of recent results about the restriction coefficients $r_\lambda^\mu$. In 2021, Heaton, Sriwongsa, and Willenbring proved the following nonvanishing result for restriction coefficients: for all positive integers $m, n > 1$ and all $\mu \vdash n$, there exists a two-row partition $\lambda = (\lambda_1, \lambda_2) \vdash mn$ such that $\lambda_1 - \lambda_2 \leq m$ and $r_\lambda^\mu > 0$ \cite{MR4244369}. In 2021, Orellana and Zabrocki introduced the \emph{irreducible character basis} $\{\tilde{s}_\lambda\}_\lambda$ of the ring of symmetric functions and used it to provide an algorithm for computing $r_\lambda^\mu$ \cite{MR4295089}. In 2024, Narayanan, Paul, Prasad, and Srivastava found a combinatorial interpretation for $r_\lambda^\mu$ in the case that $\mu$ has one column and $\lambda$ is either a hook shape or has at most two columns \cite{MR4804588}.

There are also several recent results about the inverse problem of writing the Specht module $V_\mu$ (or, more precisely, the class of $V_\mu$ in the representation ring of $\Sym_n$) as a $\Z$-linear combination of Schur modules $\mathbb{S}^\lambda \K^n$. In 2019, Assaf and Speyer proved that the coefficients $b_\lambda^\mu$ that arise in this way alternate in sign; that is, $(-1)^{|\mu| - |\lambda|} b_\lambda^\mu \geq 0$ \cite{MR4055931}. Ryba categorified this result in 2020 by finding a resolution of $V_\mu$ by direct sums of Schur modules \cite{MR4127906}.

We take the following approach to the restriction problem. Let~$\Lambda$ denote the ring of symmetric functions in the variables $x_1, x_2, x_3, \ldots$ and let $\overline{\Lambda}$ denote the ring of symmetric power series in $x_1, x_2, x_3, \ldots$. In this extended abstract, we will consider the abelian group homomorphism $\Fa \colon \Lambda \to \overline{\Lambda}$ defined on the basis $\{s_\lambda\}$ of Schur functions by \[\F{s_\lambda} = \sum_{\mu} r_{\lambda}^\mu s_\mu.\] Equivalently, $\F{s_\lambda}$ is the result of applying the Frobenius characteristic map to the representation $\bigoplus_{n} \mathbb{S}^\lambda \K^n$ of $\bigoplus_{n} \K[\Sym_n]$. For this reason, we call~$\Fa$ the \emph{Frobenius transform}. It encodes all information about all the restriction coefficients. We will use this perspective to prove that several of the restriction coefficients $r_\lambda^\mu$ vanish.

The Frobenius transform can alternatively be defined in terms of the irreducible character basis $\{\tilde{s}_\lambda\}_\lambda$ of Orellana and Zabrocki. Namely, \[\F{\tilde{s}_\lambda} = ((1 + h_1 + h_2 + \cdots)^{\perp} s_\lambda) \cdot (1 + h_1 + h_2 + \cdots)\] for all partitions $\lambda$, where $f^\perp\colon \Lambda \to \Lambda$ denotes the operator adjoint under the Hall inner product to multiplication by~$f$.

Interestingly, it is also possible to think of the Frobenius transform as a decategorification of the \emph{analytic functor} construction defined by Joyal in 1986 \cite{MR0927763}. We will not explore this perspective here; see \cite[Section~3.1]{MR4804579} for more details.

This extended abstract will be organized as follows. \Cref{sec:basics} will cover the basic properties of the Frobenius transform, several of which follow from the work of Orellana and Zabrocki. For example, for any symmetric functions $f, g$, we have $\F{fg} = \F{f} \ast \F{g}$, where~$\ast$ is the Kronecker product of symmetric functions.

In \Cref{sec:surjective}, we will define the \emph{surjective Frobenius transform} $\Fa_{\Sur} \colon \Lambda \to \Lambda$ by the equation $\FSur{f} = \F{f} / (1 + h_1 + h_2 + \cdots)$. We will discuss the basic properties of $\Fa_{\Sur}$, such as the fact that $\FSur{f}$ has the same degree and leading term as $f$. We will also introduce the inverse function $\Fa_{\Sur}^{-1}$.

In \Cref{sec:computations}, we will state a combinatorial formula for $\FSur{f}$, where~$f$ is any complete homogeneous, elementary, or power sum symmetric function. We will also state a formula for $\FSurInv{f}$, where $f$ is any elementary symmetric function.

Finally, in \Cref{sec:vanishing}, we will explain how to use the Frobenius transform to prove new and surprising results about the vanishing of the restriction coefficients $r_{\lambda}^\mu$. Namely, $r_\lambda^\mu = 0$ if $|\lambda \cap \hat{\mu}| < 2|\hat{\mu}| - |\lambda|$, where $\hat{\mu}$ is the partition formed by removing the first part of~$\mu$. Additionally, $r_\lambda^\mu = 0$ if the Young diagram of~$\mu$ contains a square of side length greater than $2^{\lambda_1 - 1}$, and this inequality is tight.

All of the results in this extended abstract are stated without proof. For proofs of all the main theorems, see \cite{MR4804579}.

\section{Preliminaries}\label{sec:preliminaries}
In this extended abstract, we will use the following terminology and notation. Definitions can be found in any standard reference on the theory of symmetric functions \cite{MR1464693}, \cite[Chapter~I]{MR3443860}, \cite[Chapter~7]{MR4621625}, or in the Preliminaries section of the author's published work \cite[Section~2]{MR4804579}.
\begin{itemize}
    \item The \emph{ring of symmetric functions} $\Lambda$ and the \emph{ring of symmetric power series} $\overline{\Lambda}$.
    \item The \emph{monomial}, \emph{elementary}, \emph{homogeneous}, \emph{power sum}, and \emph{Schur} symmetric functions, denoted $m_\lambda, e_\lambda, h_\lambda, p_\lambda, s_\lambda$ respectively.
    \item The \emph{Hall inner product} $\langle \cdot, \cdot \rangle \colon \Lambda \times \overline{\Lambda} \to \Z$.
    \item The \emph{plethysm} $f[g]$, where $f, g \in \overline{\Lambda}$.
    \item The \emph{skewing operator} $f^\perp \colon \Lambda \to \Lambda$, where $f \in \Lambda$.
    \item The \emph{Littlewood--Richardson coefficient} $c^\nu_{\lambda\mu}$.
    \item The \emph{Kronecker product} $\ast \colon \overline{\Lambda} \times \overline{\Lambda} \to \overline{\Lambda}$ and the \emph{Kronecker coefficient} $g_{\lambda\mu\nu}$.
    \item The \emph{Specht module} $V_\lambda$.
    \item The \emph{Schur functor} $\mathbb{S}^\lambda \colon \Vect_{\K} \to \Vect_{\K}$, where $\Vect_{\K}$ is the category of vector spaces over $\K$.
    \item The \emph{Frobenius characteristic map} $\ch_n$ and the \emph{Frobenius characteristic} of an $\Sym_n$-module.
\end{itemize}
We also use the notation $H$ to refer to the element $1 + h_1 + h_2 + \cdots \in \overline{\Lambda}$.

\section{The Frobenius transform: definition and basic properties}\label{sec:basics}
Let $\lambda, \mu$ be partitions, and let $n = |\mu|$. Recall from the introduction that the \emph{restriction coefficient} \[r_{\lambda}^\mu = \dim \operatorname{Hom}_{\Sym_n}(V_\mu, \mathbb{S}^\lambda \K^n)\] is defined to be the multiplicity of the Specht module $V_\mu$ in the Schur module $\mathbb{S}^\lambda \K^n$. It is a long-standing open problem to find a combinatorial interpretation of $r_\lambda^\mu$. As a potential way to approach this problem, we now define the Frobenius transform, which is the primary object of study in this extended abstract.
\begin{defi}
    The \emph{Frobenius transform} is the abelian group homomorphism $\Fa \colon \Lambda \to \overline{\Lambda}$ defined on the basis $\{s_\lambda\}$ by \[\F{s_\lambda} = \sum_\mu r_{\lambda}^\mu s_\mu,\] where the sum is over all partitions~$\mu$.
\end{defi}
\begin{rema}
By a classical result of Littlewood \cite{MR1576896}, we have \[r_\lambda^\mu = \langle s_\lambda, s_\mu[H] \rangle.\] Hence,~$\Fa$ is adjoint to plethysm by~$H = 1 + h_1 + h_2 + \cdots$ under the Hall inner product.
\end{rema}

Here is the reason for calling~$\Fa$ the Frobenius transform. Let $n \geq 0$ and let~$\lambda$ be any partition. Then $\mathbb{S}^\lambda \K^n$, considered as an $\Sym_n$-module, can be expressed as a direct sum of Specht modules:
\[\mathbb{S}^\lambda \K^n = \bigoplus_{|\mu| = n} r_{\lambda}^\mu V_\mu.\]
Taking the character of both sides and applying the Frobenius characteristic map, we obtain \begin{equation}\label{equation:fslambdan1}\ch_n(\chi_{\mathbb{S}^\lambda \K^n}) = \sum_{|\mu| = n} r_{\lambda}^\mu s_\mu .\end{equation} In other words, the degree~$n$ part of $\F{s_\lambda}$ is equal to the Frobenius characteristic of $\mathbb{S}^\lambda \K^n$.

To summarize, the restriction coefficient $r_\lambda^\mu$ can be written in terms of the Hall inner product in the following three ways: 
\[
    r_{\lambda}^\mu = \langle \F{s_\lambda}, s_\mu \rangle = \langle s_\lambda, s_\mu[H] \rangle =  \langle \ch_n(\chi_{\mathbb{S}^\lambda \K^n}), s_\mu \rangle.
\]

\begin{exam}\label{example:fer}
    Let $r \geq 0$. We will compute $\F{e_r}$. First, $e_r = s_{\lambda}$, where $\lambda = (1^r)$. Hence, for any~$n$, the degree~$n$ part of $\F{e_r}$ is the Frobenius characteristic of \[\mathbb{S}^{\lambda} \K^n = \wedge^r \K^n = \operatorname{Ind}_{\Sym_r \times \Sym_{n - r}}^{\Sym_n} (V_{(1^r)} \otimes V_{(n-r)}),\] considered as an $\Sym_n$-module. Thus, it is equal to $e_r h_{n - r}$ \cite[Proposition~7.18.2]{MR4621625}. Taking the sum over all~$n$ yields $\F{e_r} = e_r \cdot H$.
\end{exam}

Let us now describe the expansion of $\F{f}$ in the power sum basis. Following Orellana and Zabrocki \cite{MR4295089}, for any partition $\mu = (\mu_1, \ldots, \mu_\ell)$ of $n$, let us define $\Xi_\mu \in \K^n$ to be the sequence consisting of the roots of unity 
\begin{gather*}
    1, \exp\left(\frac{2 \pi i}{\mu_1}\right), \exp\left(\frac{4 \pi i}{\mu_1}\right), \ldots, \exp\left(\frac{2(\mu_1 - 1) \pi i}{\mu_1}\right), \\
    \vdots \\
    1, \exp\left(\frac{2 \pi i}{\mu_\ell}\right), \exp\left(\frac{4 \pi i}{\mu_\ell}\right), \ldots, \exp\left(\frac{2(\mu_\ell - 1) \pi i}{\mu_\ell}\right).
\end{gather*}
Then, we have the following.
\begin{prop}\label{proposition:rootsofunityformula}
    Let $f \in \Lambda$. Then \[\F{f} = \sum_{\mu} f(\Xi_\mu) \frac{p_\mu}{z_\mu}.\]
\end{prop}
This proposition can be written as \[\F{f} = \phi_0(f) + \phi_1(f) + \phi_2(f) + \cdots,\] where $\phi_n \colon \Lambda \to \Lambda$ was defined by Orellana and Zabrocki \cite[Equation~(7)]{MR4275829}. 

Another important property of the Frobenius transform is that it relates the ordinary product of symmetric functions to the Kronecker product~$\ast$.
\begin{prop}
    \label{proposition:kronecker}
    Let $f, g \in \Lambda$. Then $\F{fg} = \F{f} \ast \F{g}$.
\end{prop}
By taking both $f$ and $g$ to be Schur functions, we may think of \Cref{proposition:kronecker} as a relationship between the Littlewood--Richardson coefficients $c^\nu_{\lambda\mu}$, the Kronecker coefficients $g_{\lambda \mu \nu}$, and the restriction coefficients $r_\lambda^\mu$, as follows. This is potentially interesting because Littlewood--Richardson coefficients have a known combinatorial interpretation, but Kronecker coefficients and restriction coefficients do not. 
\begin{coro}
    \label{corollary:productcoefficients}
    Let $\lambda, \mu, \nu$ be partitions. Then
    \[\sum_{\nu'} r^{\nu}_{\nu'} c^{\nu'}_{\lambda\mu} = \sum_{\lambda', \mu'} r^{\lambda'}_{\lambda} r^{\mu'}_{\mu} g_{\lambda'\mu'\nu}.\]
\end{coro}

\section{The surjective Frobenius transform}\label{sec:surjective}
Let $f \in \Lambda$. Even though $\F{f} \in \overline{\Lambda}$ can have infinitely many nonzero coefficients, it only carries a finite amount of information in the following sense.
\begin{prop}
    \label{proposition:fsur}
    Let $f \in \Lambda$. There exists a symmetric function $\FSur{f} \in \Lambda$ such that $\F{f} = \FSur{f} \cdot H$. Moreover, $\FSur{f}$ has the same degree and leading term as~$f$.
\end{prop}
For example, in \Cref{example:fer} we showed that $\F{e_r} = e_r \cdot H$. Thus, $\FSur{e_r} = e_r$. (Note, however, that in general, $\Fa_{\Sur}$ does not preserve the property of being homogeneous.) We call $\Fa_{\Sur}$ the \emph{surjective Frobenius transform} because it is related to the category $\rmSur$ whose objects are finite sets and whose morphisms are surjective functions \cite[Proposition~3.15~proof~1]{MR4804579}.

The function $\Fa_{\Sur} \colon \Lambda \to \Lambda$ can be described explicitly as the adjoint to plethysm by $H_+ = h_1 + h_2 + h_3 + \cdots$ under the Hall inner product. It is not difficult to show that $\Fa_{\Sur}$ is invertible:
\begin{coro}
    There exists a two-sided inverse $\Fa_{\Sur}^{-1} \colon \Lambda \to \Lambda$ of $\Fa_{\Sur}$.
\end{coro}

Like the ordinary Frobenius transform, the surjective Frobenius transform can be described in terms of its matrix entries in the Schur basis.
\begin{defi}\label{def:coeffs}
Let $\lambda, \mu$ be partitions. Define the \emph{surjective restriction coefficient} $t_\lambda^\mu$ by \[t_\lambda^\mu = \langle \FSur{s_\lambda}, s_\mu\rangle\] and define the \emph{inverse surjective restriction coefficient} $u_\lambda^\mu$ by \[u_\lambda^\mu = \langle \FSurInv{s_\lambda}, s_\mu\rangle.\]
\end{defi}
By the above, we have $t_\lambda^\mu = u_\lambda^\mu = \delta_{\lambda \mu}$ for $|\mu| \geq |\lambda|$.

\begin{comment}
\section{The analytic functor construction}\label{sec:analytic-functor}
In this section, we provide an alternate perspective on the Frobenius transform based on the representation theory of categories. In particular, we explain how the Frobenius transform can be thought of as a decategorification of the \emph{analytic functor} construction defined by Joyal in 1986 \cite{MR0927763}. This subsection is not essential to the statements or proofs of our main results, so the reader may skip it. For a more complete introduction to the representation theory of categories, see Wiltshire-Gordon's 2016 PhD thesis \cite{MR3641124}.

In what follows, let $\Vect_\K$ be the category whose objects are vector spaces over~$\K$ and whose morphisms are linear transformations. (The ground field~$\K$ can be replaced by any algebraically closed field of characteristic $0$.)

\begin{defi}
    Let $\mathcal{C}$ be a category. A \emph{$\mathcal{C}$-module} (over the ground field~$\K$) is a functor $M[\bullet] \colon \mathcal{C} \to \Vect_\K$. The \emph{category of $\mathcal{C}$-modules} is the functor category $\Mod^\mathcal{C} = (\Vect_\K)^\mathcal{C}$.
    
    We say that a $\mathcal{C}$-module~$M$ is \emph{finite-dimensional} if $M[U]$ is a finite-dimensional vector space for all $U \in \operatorname{Ob}(\mathcal{C})$. The \emph{category of finite-dimensional $\mathcal{C}$-modules}, denoted $\fdMod^{\mathcal{C}}$, is the full subcategory of $\Mod^{\mathcal{C}}$ whose objects are the finite-dimensional $\mathcal{C}$-modules. This is equivalent to the functor category $(\fdVect_\K)^\mathcal{C}$, where $\fdVect_\K$ is the category of finite-dimensional vector spaces over~$\K$.

    If~$M$ and~$N$ are $\mathcal{C}$-modules, we may form the \emph{direct sum} $M \oplus N$ by $(M \oplus N)[U] = M[U] \oplus N[U]$. This makes both $\Mod^{\mathcal{C}}$ and $\fdMod^{\mathcal{C}}$ into abelian categories.

    The \emph{Grothendieck group} of $\fdMod^{\mathcal{C}}$, denoted $K(\fdMod^{\mathcal{C}})$, has the following presentation. The generators are isomorphism classes $[M]$, where $M$ is a finite-dimensional $\mathcal{C}$-module, and we have the relation $[N] = [M] + [P]$ whenever there is an exact sequence \[0 \to M \to N \to P \to 0.\]
\end{defi}

Let $\rmBij$ be the category whose objects are finite sets and whose morphisms are bijections, and let~$M$ be a $\rmBij$-module. (Previous authors have referred to $\rmBij$-modules as \emph{linear species} or \emph{tensor species} \cite{MR0927763, MR1132426}.) For $n \geq 0$, denote by $M[n]$ the vector space $M[[n]] = M[\{1, \ldots, n\}]$, which is an $\Sym_n$-module. Then~$M$ is uniquely determined, up to natural isomorphism, by the sequence $M[0], M[1], M[2], \ldots$ of symmetric group modules; namely, 
\begin{equation}\label{equation:bmodule}
M[U] = \K \Bij([n], U) \otimes_{\Sym_n} M[n]
\end{equation} for all finite sets~$U$ with $|U| = n$.
Additionally, if~$M$ is finite-dimensional, then we may form the series
\[\ch(M) = \sum_{n} \ch_n(\chi_{M[n]}) \in \overline{\Lambda},\] which we call the \emph{Frobenius characteristic of~$M$}. It also uniquely determines~$M$ up to natural isomorphism. The Frobenius characteristic map~$\ch$ descends to an isomorphism from $K(\fdMod^{\rmBij})$ to $\overline{\Lambda}$. In this way, we think of $\overline{\Lambda}$ as a decategorification of $\fdMod^{\rmBij}$.

Many well-known operations on $\overline{\Lambda}$ correspond to constructions on $\fdMod^{\rmBij}$. For example, consider the multiplication operation $\cdot \colon \overline{\Lambda} \times \overline{\Lambda} \to \overline{\Lambda}$. The corresponding construction on $\fdMod^{\rmBij}$ is the following:
\begin{defi}[{\cite[Section~4.1]{MR0927763}}]\label{definition:speciesproduct}
    Let $M, N$ be $\rmBij$-modules. Define the \emph{product} $M \cdot N$ to be the $\rmBij$-module given by \[(M \cdot N)[U] = \bigoplus_{\substack{U_1 \cup U_2 = U \\ U_1 \cap U_2 = \emptyset}} M[U_1] \otimes N[U_2].\]
\end{defi}
When we say that this construction corresponds to the multiplication operation of $\overline{\Lambda}$, we mean that the following proposition holds.
\begin{prop}[{\cite[Proposition~2.1]{MR1132426}}]
    \label{proposition:moduleproduct}
    Let~$M$,~$N$ be finite-dimensional $\rmBij$-modules. Then \[\ch(M \cdot N) = \ch(M) \ch(N).\]
\end{prop}
\end{comment}
\section{Computing the Frobenius transform}\label{sec:computations}
\subsection{Formulas for the surjective Frobenius transform}
Recall that one of the overall goals of this project is to find a combinatorial interpretation of the restriction coefficients $r_\lambda^\mu$. Since the restriction coefficients are the matrix entries of $\Fa$ in the Schur basis, this is tantamount to finding an explicit combinatorial (i.e. subtraction-free) formula that writes $\F{s_\lambda}$ as a linear combination of Schur functions.

Such a formula for $\F{s_\lambda}$ is still not known. However, \Cref{theorem:computation} below provides a combinatorial formula for $\FSur{h_\lambda}$, $\FSur{e_{\lambda}}$, and $\FSur{p_\lambda}$. Since $\F{f} = \FSur{f} \cdot H$ for all $f \in \Lambda$, \Cref{theorem:computation} can also be used to compute $\F{h_\lambda}$, $\F{e_\lambda}$, and $\F{p_\lambda}$ to any desired degree. A statement equivalent to parts \ref{item:fsurh} and \ref{item:fsure} of \Cref{theorem:computation} has appeared in the work of Orellana and Zabrocki \cite[Equation~(6)]{MR4245137}, but part \ref{item:fsurp} is new.
\begin{theorem}\label{theorem:computation}
    Let~$\lambda$ be a partition and let $\ell = \ell(\lambda)$ be its length.
    \begin{enumerate}[label=(\alph*)]
        \item \label{item:fsurh} Then \[\FSur{h_\lambda} = \sum_{M} \prod_{j \in \N^\ell} h_{M(j)},\] where the sum is over all functions $M\colon \N^\ell \to \N$ such that $M(0, \ldots, 0) = 0$ and $\sum_{j \in \N^\ell} j_i M(j) = \lambda_i$ for $i = 1, \ldots, \ell$.
        \item \label{item:fsure} Then \[\FSur{e_\lambda} = \sum_{M} \prod_{j \in \{0, 1\}^\ell}\begin{cases}
        h_{M(j)} & \mbox{if $j_1 + \cdots + j_\ell$ is even;} \\
        e_{M(j)} & \mbox{if $j_1 + \cdots + j_\ell$ is odd,}
        \end{cases}\] where the sum is over all functions $M \colon \{0, 1\}^\ell \to \N$ such that $M(0, \ldots, 0) = 0$ and $\sum_{j \in \{0, 1\}^\ell} j_i M(j) = \lambda_i$ for $i = 1, \ldots, \ell$.
        \item \label{item:fsurp} Then \[\FSur{p_\lambda} = \sum_{\pi} \prod_{U \in \pi}\left(\sum_{d \mid \gcd\{\lambda_i \colon i \in U\}} d^{|U| - 1} p_d\right),\] where the outer sum is over all partitions~$\pi$ of $\{1, \ldots, \ell\}$ into nonempty sets and the product is over all parts $U$ of $\pi$.
    \end{enumerate}
\end{theorem}
\begin{exam}\label{example:h}
    Let us use \Cref{theorem:computation}\ref{item:fsurh} to compute $\FSur{h_{2, 2}}$. First, we list all the functions $M \colon \N^2 \to \N$ such that $M(0, 0) = 0$ and $\sum_{j \in \N^2} j M(j) = (2, 2)$. There are nine such functions $M_1, \ldots, M_9$. Here are all of their nonzero values.\footnote{For readers who are familiar with the language of multisets and multiset partitions \cite{MR4295089}, it can be helpful to remember that such functions~$M$ are in bijection with multiset partitions of $\ldblbrace 1, 1, 2, 2\rdblbrace$. The multiset partition corresponding to the function~$M$ contains $M(j)$ copies of $\ldblbrace 1^{j_1}, 2^{j_2}\rdblbrace$ for all $j \in \N^2$. For example, the function $M_6$ corresponds to the multiset partition $\ldblbrace \ldblbrace 1, 1 \rdblbrace, \ldblbrace 2 \rdblbrace, \ldblbrace 2 \rdblbrace \rdblbrace \Vdash \ldblbrace 1, 1, 2, 2\rdblbrace$.}
    \begin{equation*}
        \begin{gathered}
            M_1(2, 2) = 1 \\
            M_2(1, 1) = 2 \\
            M_3(2, 1) = 1 \qquad M_3(0, 1) = 1 \\
            M_4(1, 2) = 1 \qquad M_4(1, 0) = 1 \\
            M_5(2, 0) = 1 \qquad M_5(0, 2) = 1 \\
            M_6(2, 0) = 1 \qquad M_6(0, 1) = 2 \\
            M_7(0, 2) = 1 \qquad M_7(1, 0) = 2 \\
            M_8(1, 1) = 1 \qquad M_8(1, 0) = 1 \qquad M_8(0, 1) = 1 \\
            M_9(1, 0) = 2 \qquad M_9(0, 1) = 2
        \end{gathered}
    \end{equation*}
    Thus,
    \begin{align*}
        \FSur{h_{2, 2}} &= \underbrace{h_1}_{M_1} + \underbrace{h_2}_{M_2} + \underbrace{h_1^2}_{M_3} + \underbrace{h_1^2}_{M_4} + \underbrace{h_1^2}_{M_5} + \underbrace{h_1 h_2}_{M_6} + \underbrace{h_1 h_2}_{M_7} + \underbrace{h_1^3}_{M_8} + \underbrace{h_2^2}_{M_9} \\
        &= h_1 + h_2 + 3h_{1, 1} + 2h_{2, 1} + h_{1, 1, 1} + h_{2, 2}.
    \end{align*}
\end{exam}

\begin{exam}
    Let us use \Cref{theorem:computation}\ref{item:fsure} to compute $\FSur{e_{5, 3}}$. First, we list all the functions $M \colon \{0, 1\}^2 \to \N$ such that $M(0, 0) = 0$ and $\sum_{j \in \{0, 1\}^2} j M(j) = (5, 3)$. There are four such functions $M_1, M_2, M_3, M_4$. Here are all of their nonzero values.
    \begin{equation*}
        \begin{gathered}
            M_1(1, 0) = 5 \qquad M_1(0, 1) = 3 \\
            M_2(1, 1) = 1 \qquad M_2(1, 0) = 4 \qquad M_2(0, 1) = 2 \\
            M_3(1, 1) = 2 \qquad M_3(1, 0) = 3 \qquad M_3(0, 1) = 1 \\
            M_4(1, 1) = 3 \qquad M_4(1, 0) = 2 \\
        \end{gathered}
    \end{equation*}
    Thus,
    \begin{align*}
        \FSur{e_{5, 3}} &= \underbrace{e_5 e_3}_{M_1} + \underbrace{h_1 e_4 e_2}_{M_2} + \underbrace{h_2 e_3 e_1}_{M_3} + \underbrace{h_3 e_2}_{M_4}.
    \end{align*}
\end{exam}

\begin{exam}
    Let us use \Cref{theorem:computation}\ref{item:fsurp} to compute $\FSur{p_{15, 10, 6}}$. Take $\lambda = (15, 10, 6)$ and $\ell = 3$. There are five partitions of $[\ell]$ into nonempty sets: $\{\{1, 2, 3\}\}$, $\{\{1, 2\}, \{3\}\}$, $\{\{1, 3\}, \{2\}\}$, $\{\{2, 3\}, \{1\}\}$, and $\{\{1\}, \{2\}, \{3\}\}$. Thus,
    \begin{align*}
        \FSur{p_\lambda} =& \left(\sum_{d \mid \gcd(\lambda_1, \lambda_2, \lambda_3)} d^2 p_d\right) + \left(\sum_{d \mid \gcd(\lambda_1, \lambda_2)} d p_d\right) \left(\sum_{d \mid \lambda_3} p_d\right)  \\
        &+ \left(\sum_{d \mid \gcd(\lambda_1, \lambda_3)} d p_d\right) \left(\sum_{d \mid \lambda_2} p_d\right) + \left(\sum_{d \mid \gcd(\lambda_2, \lambda_3)} d p_d\right) \left(\sum_{d \mid \lambda_1} p_d\right)\\
        &+ \left(\sum_{d \mid \lambda_1} p_d\right) \left(\sum_{d \mid \lambda_2} p_d\right) \left(\sum_{d \mid \lambda_3} p_d\right) \\
        =&\,\, p_1 + (p_1 + 5p_5)(p_1 + p_2 + p_3 + p_6) \\
        &+ (p_1 + 3p_3)(p_1 + p_2 + p_5 + p_{10}) + (p_1 + 2p_2)(p_1 + p_3 + p_5 + p_{15}) \\
        &+ (p_1 + p_3 + p_5 + p_{15})(p_1 + p_2 + p_5 + p_{10})(p_1 + p_2 + p_3 + p_6).
    \end{align*}
\end{exam}

\begin{rema}
    One consequence of \Cref{theorem:computation}\ref{item:fsurp} is that the matrix entries of $\Fa_{\Sur}$ in the power sum basis are all nonnegative integers. The same is not true of~$\Fa$; for example, $\F{1} = H = 1 + p_1 + \frac{1}{2} (p_2 + p_{1}^2) + \cdots$ certainly has some non-integer coefficients.
\end{rema}

\subsection{Formulas for the inverse surjective Frobenius transform}
Now, we turn to the inverse surjective Frobenius transform $\Fa_{\Sur}^{-1}$. It follows from a 2019 result of Assaf and Speyer \cite[Theorem~3]{MR4055931} that $\FSurInv{s_{\lambda}}$ is Schur-alternating for all partitions $\lambda$. That is, we may write \begin{equation}\label{eq:fsurinvs}\FSurInv{s_\lambda} = \sum_{\mu} u_\lambda^\mu s_\mu,\end{equation} where the coefficients $u_\lambda^\mu$ (which have appeared in \Cref{def:coeffs}) satisfy $(-1)^{|\lambda| - |\mu|}u_\lambda^\mu \geq 0$ for all partitions $\mu$ \cite[Theorem~4.2(c)]{MR4804579}. It would be interesting to find a combinatorial interpretation of $(-1)^{|\lambda| - |\mu|}u_\lambda^\mu$, or, equivalently, to find an explicit formula for $\FSurInv{s_\lambda}$ that is manifestly Schur-alternating.

Such a formula is still not known. However, \Cref{theorem:fsurinve} below provides a formula for $\FSurInv{e_\lambda}$. The formula implies that $\FSurInv{e_\lambda}$ is \emph{$e$-alternating} for all partitions $\lambda$. That is, we may write \begin{equation}\label{eq:fsurinve}\FSurInv{e_\lambda} = \sum_{\mu} d_\lambda^\mu e_\mu,\end{equation} where the coefficients $d_\lambda^\mu$ satisfy the property that $(-1)^{|\lambda| - |\mu|}d_\lambda^\mu$ is a nonnegative integer for all $\mu$. Moreover, the formula provides an explicit combinatorial intepretation of the number $(-1)^{|\lambda| - |\mu|}d_\lambda^\mu$.

Before we proceed to the statement of \Cref{theorem:fsurinve}, let us recall some definitions from the combinatorics of words. For a more complete introduction, see \cite[Chapter~5]{MR1475463}.

\begin{defi}
    Let~$A$ be a set. A \emph{word} over the alphabet~$A$ is a sequence $w = w_1 \cdots w_n$ with $w_1, \ldots, w_n \in A$. Given a letter $a \in A$, we write $m_a(w)$ to denote the number of times the letter~$a$ appears in~$w$.
\end{defi}
\begin{defi}[\cite{MR0067884}]
    Let~$A$ be a totally ordered set. We say that a nonempty word $w = w_1 \cdots w_n$ over the alphabet~$A$ is a \emph{Lyndon word} if it is lexicographically less than its suffix $w_i \cdots w_n$ for $i = 2, \ldots, n$. Let $\Lyndon(A)$ be the set of all Lyndon words over the alphabet~$A$.
\end{defi}

\begin{theorem}[{Chen--Fox--Lyndon Theorem \cite{MR0102539}}]\label{theorem:chenfoxlyndon}
    Let~$A$ be a totally ordered set. Any word~$w$ over the alphabet~$A$ has a unique \emph{Lyndon factorization}; that is, an expression as a (lexicographically) non-increasing concatenation of Lyndon words.
\end{theorem}
\begin{defi}
    Let~$w$ be a word over a totally ordered alphabet. Define $\pi(w)$ to be the partition obtained by listing the number of times each Lyndon word appears in the Lyndon factorization of~$w$, and then sorting the resulting positive numbers in decreasing order.
\end{defi}
\begin{exam}
    If $A = \{1, 2\}$ and $w = 21212121111$, then the Lyndon factorization of~$w$ is $w = (2)(12)(12)(12)(1)(1)(1)(1)$. The Lyndon words appearing in this factorization are $2$, $12$, and $1$, which appear once, three times, and four times, respectively, so $\pi(w) = (4, 3, 1)$.
\end{exam}

We are now ready to state our formula for $\FSurInv{e_\lambda}$.
\begin{theorem}\label{theorem:fsurinve}
    Let $\lambda = (\lambda_1, \ldots, \lambda_\ell)$ be a sequence of nonnegative integers \textup{(}not necessarily weakly decreasing\textup{)}. Then \[\FSurInv{e_\lambda} = \sum_{w \in W} (-1)^{|\lambda| - |\pi(w)|} e_{\pi(w)},\] where~$W$ is the set of all words~$w$ over $[\ell]$ such that $m_i(w) = \lambda_i$ for all $i \in [\ell]$.
\end{theorem}

In other words, if $d_\lambda^\mu$ is the coefficient of $e_\mu$ in $\FSurInv{e_\lambda}$ as in \eqref{eq:fsurinve}, then $(-1)^{|\lambda| - |\mu|}d_\lambda^\mu$ is the number of words $w$ such that $m_i(w) = \lambda_i$ for all $i \in [\ell]$ and $\pi(w) = \mu$.

\begin{exam}
    Let us use \Cref{theorem:fsurinve} to compute $\FSurInv{e_{3, 2}}$. In \Cref{figure:lyndon-table}, we list all the words $w$ over $[2]$ such that $m_i(w) = \lambda_i$ for all $i \in [2]$. These are exactly the words that consist of three $1$'s and two $2$'s. For each such word $w$, we record the Lyndon factorization of $w$ and the partition $\pi(w)$.
    \begin{figure}
        \begin{center}
        \begin{tabular}{ccc}
            $w$ & Lyndon factorization & $\pi(w)$ \\
            \hline
            $11122$ & $(11122)$ & $(1)$ \\
            $11212$ & $(11212)$ & $(1)$ \\
            $11221$ & $(1122)(1)$ & $(1, 1)$ \\
            $12112$ & $(12)(112)$ & $(1, 1)$ \\
            $12121$ & $(12)(12)(1)$ & $(2, 1)$ \\
            $12211$ & $(122)(1)(1)$ & $(2, 1)$ \\
            $21112$ & $(2)(1112)$ & $(1, 1)$ \\
            $21121$ & $(2)(112)(1)$ & $(1, 1, 1)$ \\
            $21211$ & $(2) (12)(1)(1)$ & $(2, 1, 1)$ \\
            $22111$ & $(2)(2)(1)(1)(1)$ & $(3, 2)$
        \end{tabular}%
        \caption{For each word $w$ over the alphabet $[2]$ that consists of three $1$'s and two $2$'s, we compute the Lyndon factorization of $w$ and the partition $\pi(w)$.}\label{figure:lyndon-table}%
        \end{center}%
    \end{figure}
    Reading from the table, we find that
    \[\FSurInv{e_{3,2}} = 2e_1 - 3e_{1, 1} + e_{1, 1, 1} + 2e_{2, 1} - e_{2, 1, 1} + e_{3, 2}.\]
\end{exam}

We also have a similar alternating formula for $\FSurInv{h_\lambda}$ \cite[Theorem~7.7(b)]{MR4804579}, but it is more complicated and we will not reproduce it here.

\section{Vanishing of restriction coefficients}\label{sec:vanishing}

One important question about $r_\lambda^\mu$ is the following: under what circumstances do we have $r_{\lambda}^\mu = 0$? Using the Frobenius transform, we can partially answer this question. For any partition~$\mu$, let $D(\mu)$ denote the size of the Durfee square of~$\mu$. That is, $D(\mu)$ is the largest integer~$d$ such that $\mu_{d} \geq d$ \cite[Chapter~8]{MR2122332}.

\begin{theorem}\label{theorem:durfeebound}
    Let~$\mu$ be a partition and let $k \geq 1$ be an integer. The following are equivalent:
    \begin{enumerate}[label=(\Alph*)]
        \item There exists a partition~$\lambda$ such that $\lambda_1 \leq k$ and $r_{\lambda}^\mu > 0$.
        \item $D(\mu) \leq 2^{k - 1}$.
    \end{enumerate}
\end{theorem}
In particular, if $\lambda$ and $\mu$ are partitions with $D(\mu) > 2^{\lambda_1 - 1}$, then $r_\lambda^\mu = 0$. 

We also have the following. For any partitions $\lambda, \mu$, let $\lambda \cap \mu$ denote the partition whose Young diagram is the intersection of the Young diagrams of $\mu$ and $\lambda$. Explicitly, $\ell(\lambda \cap \mu) = \min(\ell(\lambda), \ell(\mu))$ and $(\lambda \cap \mu)_i = \min(\lambda_i, \mu_i)$ for all $i$.

\begin{theorem}\label{theorem:surjectivevanishing}
    Let $\lambda, \mu$ be partitions. If the surjective restriction coefficient $t_\lambda^\mu$ does not vanish, then $|\lambda \cap \mu| \geq 2 |\mu| - |\lambda|$.
\end{theorem}

\begin{theorem}\label{theorem:vanishing}
    Let $\lambda, \mu$ be partitions. If the restriction coefficient $r_\lambda^\mu$ does not vanish, then $|\lambda \cap \hat \mu| \geq 2|\hat \mu| - |\lambda|$, where $\hat \mu = (\mu_2, \ldots, \mu_{\ell(\mu)})$ is the partition formed by removing the first part of~$\mu$.
\end{theorem}
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\acknowledgements{The author thanks Mike Zabrocki, Katherine Tung, and the FPSAC 2025 Program Committee for providing helpful feedback on an earlier version of this extended abstract.}

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